Izbirno tekmovanje (selection round) 2003 · Lemma

1Number theoryFree answerFind every natural number with the following property: when the sum of its digits is added to the number itself, the result is \(313\).2GeometryFree answerLet \(M\) be the midpoint of the side \(BC\) and \(N\) the midpoint of the side \(CD\) of a rectangle \(ABCD\). Determine the ratio of the side lengths of \(ABCD\), given that \(AMN\) is a right triangle …3AlgebraProofLet \(a\), \(b\), \(c\) be positive numbers with \(a^2 + c^2 = 2b^2\). Prove that \[ \frac{2}{a+c} = \frac{1}{a+b} + \frac{1}{b+c} . \]4GeometryFree answerGrandmother cut a round pizza into \(6\) equilateral triangles and \(6\) circular segments, as in the picture. Each of her \(6\) grandchildren ate one triangle. The grandchildren do not like the crust, …5Number theoryFree answerLet \(a\) and \(b\) be natural numbers, both greater than \(1\), such that \[ \sqrt{a\sqrt{a\sqrt{a}}} = b . \] What is the smallest possible value of the sum \(a + b\)?

Work through the paper in order - each problem opens in the workspace with this paper as its trail.